Andrew B. Wright, Ph. D., SM ’88
On one of my robotics projects, I ran into a need to determine the moment of inertia of my wheel assemblies. I used a measurement technique that I had been taught in school.
I created a disk of precisely machined radius and attached it to the wheel axle. I wrapped a string around the disk and attached a bottle with water in it. I dropped the bottle and measured the angular wheel speed with an encoder. I plotted this speed and measured the slope of the linear region. That slope is the acceleration in the linear region.
I weighed the bottle. The wheel radius and bottle weight provided the constant torque.
I used different weights, by filling the bottle, and measured moment of inertia at different points. However, when I plotted the graph, there was an offset near zero.
When I did macro-scale experiments on the entire robot, my values of moment of inertia didn’t seem to track either. I eventually determined that my MoI was probably fine, but that there was weight-dependent friction in my system. The wheel bearings were cheap delrin plastic bearings, so this should not have come as a shock.
I redesigned the system to use bearings, which made the problem go away. However, it sparked some thoughts about how to measure weight-dependent friction in my drive system. And, while I presume that someone else has long since figured this out, I might as well play around with this experiment to see where it goes.
Here’s a picture describing the set up. I designed a wheel (originally machined out of aluminum, but in the present experiment, it will be 3D printed from ABS).
The wheel has a 1/8″ square hole to turn the axle. It has grooves to capture the string. I has a lip on either side to keep the string from riding up and off the wheel.
Rather than hanging one weight from one end of the string, weights are placed on both sides of the wheel. In this way, the sum of the weights determines the total weight on the wheel bearing, while the difference in weights determines the torque. This allows both torque and weight to be varied, and it allows graphs of torque versus acceleration to be plotted for different weights.
In the absence of friction, all the curves would project to the origin (ie. zero torque yields zero acceleration). For constant friction, all the curves would trace to an offset (torque below the offset would yield zero acceleration). In the case of weight dependent friction, the offsets would be different for different load (i.e., you could construct a curve of offset versus weight). In the case of speed dependent friction, the angular velocity versus time curve would not have a linear portion, and the measurement of acceleration would be tricky. Such a measurement would rely on sinusoidal oscillation of the system.
Until proven otherwise, I’m going to presume constant friction for bearings and weight dependent friction for the bushings.
The speed measurement system is documented in this post.
ASTM standards on friction measurement